Yes! Absolutely you can have this done. The quality of work may vary depending on the level of experience of the shop you commission to do the work but you could end up with something like this, if you want.
I would not remove the entire exhaust and take it to a shop.
How are they going to know where the exhaust should run without having the engine bay and frame available to measure everything up to ensure it fits into the engine bay correctly?
Running the engine with no exhaust is not good for your exhaust valves.
Exhaust fumes from your engine bay will get into the cabin and you will be breathing in some very noxious fumes that kill living organisms, that being you and whomever else get's in.
Keep the header/manifold
If you can use your current exhaust manifold that comes off your cylinder head, that's the most cost effective and reliable way. You don't want anyone fabricating that piece. You would then cut off the beginning of your exhaust pipe that bolts up to the manifold to use that. Your fabricator can then weld tubing that is the same as the current diameter of your exhaust pipe to that connector piece so it can bolt up to your manifold with the right connection.
Add in a muffler
You can then weld a muffler into the system. It's level of restriction may force you to tune your carburetor to match the restriction in the muffler. More backpressure will richen your system. Less will lean it out. You don't want to run rich or lean.
Rich wastes fuel
Lean runs hot, more oxygen, and can damage your exhaust valves and 'burn' them.
Here's a QA on burnt valves
Yes. You can do it.
Put a muffler in it too, you need the restriction and the back pressure.
If you love the planet, put a catalytic converter in it.
You will probably have to tune your air fuel ratio to get things proper after you install your exhaust.
Take the whole car in when you do it so they can fabricate it properly and run the exhaust to match the current fitment.